Resolution of an Open Problem on Quasi-Cyclic Codes over ℤ4 and New Quaternary Linear Codes
Abstract
Given a cyclic code Cg = ⟨ g(x) ⟩ of odd length m over ℤ4, one common way to build a quasi-cyclic (QC) code is to pick f1, …, fℓ ∈ ℤ4[x] and let C = ⟨ (f1 g, …, fℓ g) ⟩. Because ℤ4 is not a field, the type of Cg (4k12k2) is not necessarily inherited by C. Determining conditions under which the type of Cg is inherited by C was posed as an open problem recently in \cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single fi to be coprime to xm−1 over 𝔽2, the other only requires the fi to be jointly coprime to it. Neither condition is necessary in general. Using the fact that xm−1 is squarefree for odd m, we decompose ℤ4[x]/⟨ xm−1⟩ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on f1, …, fℓ that is both necessary and sufficient for C to match the type of Cg. This condition depends only on the irreducible factors of xm−1 where g does not already vanish. This also yields a simple test for when C is a free ℤ4-module. Finally, we report many new QC codes over ℤ4, found by computer searches using Magma software~\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.
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BibTeX
@misc{aydin2026resolution,
title = {{Resolution of an Open Problem on Quasi-Cyclic Codes over $\mathbbZ_4$ and New Quaternary Linear Codes}},
author = {Nuh Aydin and Aditya Tyagi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21601},
eprint = {2609.21601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21601},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Resolution of an Open Problem on Quasi-Cyclic Codes over ℤ_4 and New Quaternary Linear Codes
AU - Nuh Aydin
AU - Aditya Tyagi
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21601
UR - https://arxiv.org/abs/2609.21601
AB - Given a cyclic code C_g = ⟨ g(x) ⟩ of odd length m over ℤ_4, one common way to build a quasi-cyclic (QC) code is to pick f_1, …, f_ℓ ∈ ℤ_4[x] and let C = ⟨ (f_1 g, …, f_ℓ g) ⟩. Because ℤ_4 is not a field, the type of C_g (4^k_12^k_2) is not necessarily inherited by C. Determining conditions under which the type of C_g is inherited by C was posed as an open problem recently in \cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single f_i to be coprime to x^m−1 over 𝔽_2, the other only requires the f_i to be jointly coprime to it. Neither condition is necessary in general. Using the fact that x^m−1 is squarefree for odd m, we decompose ℤ_4[x]/⟨ x^m−1⟩ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on f_1, …, f_ℓ that is both necessary and sufficient for C to match the type of C_g. This condition depends only on the irreducible factors of x^m−1 where g does not already vanish. This also yields a simple test for when C is a free ℤ_4-module. Finally, we report many new QC codes over ℤ_4, found by computer searches using Magma software~\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.
ER -